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Full text of "Treatise On Analysis Vol-Ii"

162       XIII    INTEGRATION
lim NPOJ = 0 (13.8.1). This shows that the sequence (/„) is a Cauchy

»-* oo

sequence in Lg , and converges to / by virtue of (ii) and the hypotheses.

A sequence (/„) in &i which converges to /e «^g with respect to the
topology defined by the seminorm Np is said to converge in the mean to / when
p = 1, and to converge in square mean to/ when p = 2. It can happen that
/„ tends to / in this sense but that the sequence (/„(») does not converge
in R for any x e X (Problem 1).

(1 3.1 1 .5) Let $ c: g g be a dense set of functions (with respect to the topology
defined by the seminorm Np). Then, for every function f e <£&, there exists a
sequence of functions gne$ such that the sequence (gn(x)) converges almost
everywhere tof(x) and such that gn tends to fin JafJ .

This follows immediately from (13.11.4) and the fact that in the metric
space L£ there exists a sequence (#„) of classes of functions belonging to $ ,
with /as limit (3.13.13).

This result applies in particular when $ = ^TR(X) :

(13.11.6)    (i)   The subspace jf R(X) is dense in JS?H(X, u) (p = 1 , 2).

(ii) For each compact subset K o/X, the topology induced on tf R(X; K)
by the topology of =£fR(X, u) is coarser than the topology defined by the
norm \\f\\.

(in)   The spaces JSf^X, u) are separable (p = 1, 2).

Assertion (i) has already been proved for p — 1 (13.7.2). To prove it
for p = 2, it is enough to show that every function / ^ 0 in <£ R lies in the
closure of <?fR(X). Now by (13.7.2), given any e > 0, there exists a function
u e <2fR(X) such that N^/2 - u) <J e, and since |/2 - u+\ <; |/2 — u\ we may
assume that u ^ 0, hence that u = v2 with v e Jf R(X) and v S> 0. But then
|/— v\2 ^ |/2 — u2|, and consequently

(N2(/- i;))2 = IW- v\2) ^ Nid/2 - v2\) g a.
To prove (ii) it is enough to observe that for any # e ^ R(X; K) we have

Finally, let (Kn) be an increasing sequence of compact sets which cover
X and are such that Jfm(X) is the union of the «#~R(X;Kn) (3.18.3). By
(7.4.4) and (3.10.9), for each n there exists a dense sequence (gmn)m*i in the
Banach space «2fR(X; Kn). From (i) and (ii) above, the double sequence
(0w)miu,ni=i is dense in JSP{(X, u).mit of